THEORETICAL ASPECTS OF VIBRATIONAL SPECTROSCOPY OF CONDENSED SYSTEMS WITH IMPURITY PARTICLES Текст научной статьи по специальности «Физика»

@

CHEMICAL PROBLEMS 2023 no. 3 (21) ISSN 2221-8688

211

UDC 535.32

THEORETICAL ASPECTS OF VIBRATIONAL SPECTROSCOPY OF CONDENSED

SYSTEMS WITH IMPURITY PARTICLES

T. Marsagishvili, M. Machavariani

Ivane Javakhishvili Tbilisi State University, R. Agladze Institute of Inorganic Chemistry and Electrochemistry. Mindeli str. 11, 0186, Tbilisi, Georgia, e-mail: tamaz.marsagishvili@gmail.com

Received 12.05.2023 Accepted 28.07.2023

Abstract: Some problems of vibrational spectroscopy of particles in condensed systems are considered in this work. One of the aspects of theoretical research is the study of the vibrational properties of individual particles in view of the nano-dimension of the molecules of the condensed system surrounding the particle. Using the apparatus of temperature, Green functions of the operators ofpolarization of condensed systems, two main mechanisms of influence on impurity particles from the medium, solvation and fluctuation, are distinguished. Theoretical results are obtained within the framework of these two mechanisms for calculating changes in the vibrational spectrum of individual particles. The theoretical results are used to analyze the experimental data on the vibrational spectra of the N2O molecule in polar solvents: methanol, ethyl alcohol, acetone, and 1,2-dichloroethane.

Keywords: vibrational spectroscopy, condensed systems, solvation, impurity particles, spatial dispersion, Green functions.

DOI: 10.32737/2221-8688-2023-3-211-220

Introduction

The modern development of studies in the complex condensed systems is increasingly approaching the level of the nanoscale. One of the aspects of research is the study of the vibrational properties of individual particles, taking into account the nano-dimension of the molecules of the condensed system surrounding the particle [1-3]. The condensed medium, depending on the polarity, significantly affects the vibrational spectrum of individual particles, causes a shift and broadening of the absorption peaks, and can even lead to the dissociation of particles into individual fragments. In this case, the influence of the condensed medium on the vibrational spectrum of impurity particles mainly occurs through two mechanisms of interaction - solvation and fluctuation mechanisms. The first of these two mechanisms assumes the solvation of the impurity molecule by the molecules of the medium; the second assumes the interaction of intramolecular

vibrations of the impurity with fluctuations of the medium polarization.

Naturally, there is no exact theoretical solution to this kind of problem [4-13]. Real approximate methods assume the use of such models when describing the properties of a condensed medium, which allow one to describe the influence of the medium on the characteristics of an individual impurity particle with the help of measured quantities. The main effect of the influence of a non-regular polar condensed medium in calculating the vibrational spectrum of an impurity can be described using the temperature Green functions of the system's polarization operators. This makes it possible to take into account the effects of frequency and spatial dispersion of a condensed medium within the framework of various models when calculating the spectral characteristics of impurity particles.

www.chemprob.org

CHEMICAL PROBLEMS 2023 no. 3 (21)

1. Hamiltonian model of impurity particle in Hamiltonian - an impurity particle in a a condensed medium, solvation condensed medium can be represented as: mechanism of interaction of an impurity particle with a polar medium.

H = H0 + £asQs° + XhstQsV (1)

Here (and below) summation over repeated indices is assumed, H0 is the Hamiltonian of the impurity particle in vacuum, Qs0 is the normal coordinate of the s-th intramolecular vibration of the impurity particle (it is assumed that we

are working in a system in which the normal value of the coordinate Qs0 is zero), the elements as and hst are determined by the relations:

Here, E is the strength of the electric field medium polarization operators P(r,t) at a finite created by the polarized impurity particle, temperature T [3]: is the Green function of the

In general form, the calculation of the characteristic parameters of an impurity particle in a polar medium is rather cumbersome. It is more convenient to introduce some characteristic model functions, which can be used to study various aspects of the influence of the polar medium on the characteristic impurity parameters. For such problems, modeling of the Green functions of the medium polarization

Where d is the dipole moment of the particle, ûùe0 is the frequency of intramolecular oscillations of the particle in the gas phase (in vacuum), qs the normal coordinate of the s-th

As can be seen from these formulas, taking into account the solvation mechanism of the interaction of an impurity particle with a polar medium led to an increase in the equilibrium length of the corresponding chemical bond

operators at a finite temperature is convenient.

To obtain evaluative values for as and hst, one can perform calculations for a simple model of a homogeneous isotropic medium, not considering the effects of spatial dispersion. For the electric field strength of an impurity particle in the dipole approximation for a symmetric particle of radius ro, we obtain:

(5)

intramolecular vibration of the impurity, with an equilibrium value equal to zero. For a particle with one optically active degree of freedom, we obtain

(6)

(parameter a) and to a decrease in the frequency of the corresponding intramolecular vibration (parameter h).

2. Hamiltonian model of a condensed medium with impurity particles, fluctuation

&ncojtoao / 3d '

\dqa

mechanism of interaction of an impurity particle with a polar medium.

Taking into accountthe interaction of medium polarization fluctuations with intramolecular vibrations of impurity particles can be essential for the processes of intramolecular relaxation of polyatomic impurity particles. Information on the effect of medium polarization fluctuations on the frequencies of intramolecular vibrations of impurity particles can be obtained from an analysis of the frequency spectrum of the Green

Where G°u(ton) is the temperature of the Green function of operators of a solvated polarized oscillator, Uuw(<jon) is the renormalized interaction of intramolecular vibrations of a

Obviously, the real frequency spectrum of an impurity particle will be determined by the vacuum frequency spectrum of the particle, so by the frequency spectrum of fluctuations of the medium and the interaction of intramolecular

function of the operators of normal coordinates of intramolecular vibrations of particles. When it comes to uncharged impurity particles, the interaction of fluctuations of the medium polarization with the static field of the impurity can be neglected. In this case, for the Green functions of the operators of normal coordinates of intramolecular vibrations Q of an impurity, a system of linear inhomogeneous algebraic equations of the following form can be obtained:

(7)

particle, taking into account the interaction of vibrations through a polar medium. In formula (7) the renormalized interaction has the form:

(8)

vibrations of the particle with fluctuations of the medium polarization.

In the factorization approximation, the Green function of the operators of medium polarization can be represented as:

(¿>n= 2irnkT, n = 0, ±1, ±2, ±3,

£ü„ =2irnkT, n = 0, ±1, ±2, ±3,....

SikO'"J = fOJSikO'1"') . (9)

Uuv(con)= f(con)A^v, (10)

The parameter Auv depends on the shape of the dispersion of the condensed medium, particle, on the derivatives of the dipole moment For spherically symmetric particles placed

of the particle with respect to the normal in a local homogeneous medium, the parameter

coordinates of the intramolecular vibrations of '"v is equal to the impurity, and on the effects of the spatial

Where ro is the radius of the particle.

The simplest model for taking into account the

effects of spatial dispersion of a condensed

medium is the function f(uj.ri):

pole approximation for the

(13)

Where (jok are the natural vibration frequencies of the medium, which can be complex, m is a number of poles, afe are the intensities of the absorption peaks.

3. Green's function of operators of normal coordinates of intramolecular vibrations of an impurity particle with one optically active degree of freedom.

Above, a model was accepted within the

ggq= OI +

framework of which the analytical form of the renormalized interaction is known. To analyze the vibration frequency spectrum of an impurity particle, it is necessary to carry out an analytic continuation from the discrete points cjow=

2nnkT to the complex frequency plane.

As a result, for a particle with one optically active frequency, we obtain for the Green function

UOJ)-1 (14)

when using the pole approximation for renormalized interaction U in the form (13), the vibration spectrum is determined from the equation

where ros is the frequency of the intramolecular vibration of the impurity particle. For the Laplacian image of the time retarded function of the operators of normal coordinates of intramolecular vibrations of an impurity,

+ p2) n\=1(.o>i-p) + 0.

A special case of the pole approximation of the function f(co) is the Debye function:

Yo+P

And for the gqq(p) function at that, we have Where k is the bond strength coefficient

Yo K

K =

ACq 4rr

(15)

(16)

(17)

(18)

co is a dimensionless constant equal to the difference between the reciprocal values of the dielectric constant of the medium to the right and left of the corresponding absorption peak.

F(p) = (p2 + uBXp+ Y„)

The frequency absorption spectrum of the system is determined from the solution of the equation

Yo^

:k= 0.

(19)

This equation always has one real root, and the other two are either real or complex conjugates, depending on the parameters in the equation (19).

The real root of this equation p=po means the dissociation of the molecule if the polar medium is a liquid, or the transition of the molecule to another conformational state if the medium is a solid.

As the analysis of equation (19) shows, if the

bond strength coefficient k <1, then for the two roots of equation (19) pi and p2 the relation RePi < C, Rep2 < Ois valid.

In this case, the time retarded Green function of the operators of normal coordinates of intramolecular vibrations of the impurity for the Debye approximation of the Green function of the operators of fluctuations of the medium polarization has the form

+

gqq(0 =

^B(YI} + PZ) (P2 - PoXPs ~ Pi)

eODC;

"a^Yo+Pp)

(Po-PiXp0-p£) exp(p2t))

exp(p(jt) +

"3(y0+Pi) (Pi-p0)(pi-p2)

exp(pit)

(20)

The first term on the right side of the relation characterizes the process of pure damping of oscillations in time, the next two terms are either oscillations (at k = 0), or damping with oscillations, or pure damping. Decays with

Where ror is the characteristic frequency of medium oscillations, y is the attenuation of medium oscillations.

U(ton) = K

oscillations have a physical meaning at k « 1.

For the resonance approximation of the Green function of the operators of fluctuations of the polarization of the medium, we have

(21)

OJ1+ OJr+i y *

The renormalized interaction has the form as follows:

(22)

On+Y^+c^

The Laplasian image of the Green function of intramolecular vibrations of an impurity has the the operators of normal coordinates of form

Gqq - -

(y +<oi )((P+ y)2 ~ (to* + y2 )) '

Where the Green function of poles are determined from the equation

(23)

(24)

It can be seen from the general analysis that this function can have either two real roots and two -complex conjugates or all roots complex (pair wise) conjugates. The study of the equation shows that if k <1, then the real roots of the equation (24) are always negative, and the real parts of the complex conjugate roots vary from -y to 0. Thus, the bond strength coefficient

0.= + tjOg).

For the complex conjugate values of the roots of equation (24) we have

should not exceed the critical value k =1. Equation (24) has real roots at the values of the bond strength coefficient k* < k < 1, where k* is the value at which the values of the real roots coincide, i.e. = p2 = a. For highly resonant media where the condition y « ojr is satisfied, the value of a is equal to

Pa,4

(Of +co|

In this case, the value of k* is equal to

+ (1 - Y2

0 ■

K* = 1

to* +2 to*

,Z ,Z , , Z

When a strong connection with the medium is 1 approximate value of only one real root - k « 1 for arbitrary values of y, one can find an

Pi= -(1-

If the medium is described by undamped occurs in the system, at which oscillations with y = 0, then the "beating" effect

(25)

(26)

(27)

(28)

PM = ± + <*l + (.(.<4 - O2 + 4kc^)m)m

Vz

(29)

It is easy to see that in the limit k = 0 we obtain with the characteristic frequencies of the

the frequencies of non-interacting oscillators impurity particle and the medium:

Similarly to the calculations performed, one can particle has two optically active degrees of

also analyze systems in which an impurity freedom.

4. Vibration spectra of a molecule N2O in polar solvents: methyl alcohol, ethyl alcohol, acetone, 1.2-dichloroethane.

During the transition from the vacuum state to the condensed phase, the vibration spectra of the molecule N2O undergo significant changes - there is a shift in the maximum of the absorption peak, broadening of the absorption peak, redistribution of intensities between different absorption peaks [14, 15]. Several solvents were chosen for quantitative processing of the experimental data. Methyl and ethyl alcohols, for the reason that they are the same type of solvents, however, the shifts in the absorption peak of the frequencies of the valence vibrations of the molecule N2O occur in different directions. Such effects are observed in

Where oj3 is the frequency of the vibration level, k is a phenomenological parameter, the value of which can be estimated from

U(to) = KS I*

acetone - the frequency shifts towards an

increase, and in 1.2 dichloroethane - the

frequency decreases.

The frequency of the valence vibration of a

bond v3 of N2O molecule in vacuum is 2223 cm" 1

Obviously, in this case, the frequency of the main level will be 1111.5 cm-1, and the frequency of the first excited level will be 3334.5 cm-1.

It is essential to take into account the effects of frequency and spatial dispersions of solvents phenomenologically.

The change in the square of the frequency of the valence vibration of the impurity, caused by the solvation of the impurity by the molecules of the medium, will take the form:

(31)

experimental data.

The frequency dependence of the renormalized interaction can be presented in the form

--(32)

(u-^ + ïf

where 5 is a dimensionless variable parameter, the values of the parameters ui, yi and roi can be determined from the absorption spectra of the

solvents [16]. Measurements were carried out on a Specord 75 IR infrared spectrophotometer (Table 1-4).

Table 1. Spectral parameters of the IR absorption spectrum o methyl alcohol

Table 2. Spectral parameters of the IR absorption spectrum of ethyl alcohol

£Oi Yi

1030 60 0.85

1120 60 0.25

2850 60 0.75

3335 400 0.90

Table 3. Spectral parameters of the IR absorption spectrum of Acetone

0Jt Yi «É

1100 30 0.23

1220 25 0.85

1365 35 0.83

3000 100 0.32

Yi «i

1050 60 0.95

1085 60 0.80

2750 200 0.75

3325 380 0.90

Table 4. Spectral parameters of the IR absorption spectrum of 1,2- dichloroethane

£Oi Yi

1015 25 0.45

1230 25 0.73

1315 25 0.70

2950 95 0.35

The energies of the vibration levels of an assumption that the valence vibration does not impurity can be easily calculated under the interact through the medium. In this case, to

calculate the energy of vibration levels of the particle, we obtain the following equation:

Based on this level, the values of the main level (frequency ci>0) and the first excited

level (frequency c^) of the bond N - O are calculated at various values of the parameters of the bond strength with the medium k and the dielectric properties of the solvent 5 (see Table5).

According to this Table, the solvation and fluctuation mechanisms of the interaction of a molecule N2O with solvents leads to a shift of the ground and excited vibration levels towards higher energies. The energy of the vibration transition from the ground level to the energy level, in which all vibrations of the molecule, except v3, are in the ground state, the valence vibration of the N-O bond in the first excited

state was determined as the difference

When choosing the parameters k and 5, it must be taken into account that the parameter k is dimensionless, see formula (31). For this reason, it is natural that when calculating for the frequency^o, the parameter k will be 3 times larger than for oe^.When it comes to the parameter 5 in acetone and 1,2-dichloroethane it can be chosen to be approximately the same for the frequencies oj,-, and oi^ since the absorption spectra of these solvents are separate peaks against a uniform background (input of the parameter 5 is associated with the presence of this background).

Table 5. Vibration bond levels of the molecule of the ground and first excited state in methyl, ethyl

alcohols, acetone and 1.2-dichloroethane.

©0

Solvent K 50 K 51

0.1 0.3 0.5 1.0 0.1 0.3 0.5 1.0

Methyl 0.02 1105 1104 1103 1099 0.01 3324 3321 3318 3309

alcohol 0.03 1102 1100 1098 1093 0.015 3319 3314 3309 3296

0.02 3314 3308 3301 3284

Ethyl 0.02 1104 1101 1098 1089 0.01 3325 3321 3318 3310

alcohol 0.03 1101 1096 1091 1076 0.015 3320 3315 3310 3297

0.02 3314 3308 3301 3285

Acetone 0.02 1106 1105 1104 1103 0.01 3322 3322 3322 3321

0.03 1103 1102 1102 1098 0.015 3318 3317 3317 3317

0.02 3326 3326 3326 3326

1,2- 0.02 1106 1106 1106 1105 0.01 3326 3326 3326 3325

dichloroethane 0.03 1103 1103 1103 1102 0.015 3322 3322 3322 3322

0.02 3318 3318 3318 3318

For methyl and ethyl alcohols, the IR spectra have a more complex structure with an uneven background, and when calculating Q = — cjOq, the parameters Sa,, Si for and oi1

can have different values for the ground and first excited energy levels.

The best agreement between the calculated frequency shift AQcaic and the experimental data AQexp for all solvents was obtained at 0.003 for and = 0.001 for the level For parameters 5 - choice from a wider range (Table 6).

Table 6. Theoretical and experimental values of the shift of absorption peaks of the N2O molecule in methyl, ethyl alcohols, acetone and 1.2-dichloroethane.

Solvent Ko ft! So Si Q cm 1 AQcalc, cm"1 AQexp , cm"1

Methyl 0.03 0.01 0.3 0.1 2224 -1 -0.9

alcohol 0.5 0.3 2223 0

0.03 0.01 0.1 0.3 2220 3 0.8

0.3 0.5 2222 1

Ethyl 0.03 0.01 0.1 0.1 2223 0 -2.6

alcohol 0.3 0.3 2224 -1

0.5 0.5 2224 -1

1.0 1.0 2228 -5

1,2- 0.03 0.01 0.1 0.1 2223 0 0.9

dichloroethane 0.3 0.3 2223 0

0.5 0.5 2223 0

1.0 1.0 2223 0

Conclusion

The mechanisms of the influence of a condensed medium on the vibration spectrum of individual impurity particles are considered. It shows that the condensed medium causes a shift and broadening of the absorption peaks of impurity particles and can lead to the dissociation of particles into separate fragments.

It should be noted that the effect of a condensed medium on the vibration spectrum of impurity particles occurs through two interaction mechanisms - solvation and fluctuation. Each mechanism was studied in

detail, and analytical results were obtained that allow one to take into account the effects of the solvation of an impurity molecule by the molecules of the medium, as well as the interaction of intramolecular vibrations of the impurity with fluctuations of the medium polarization.

Theoretical results are applied to analyze the vibration spectra of the N2O molecule in polar solvents: methanol, ethyl alcohol, acetone, 1.2-dichloroethane.

References

1. Deffeys K., Deffeys S. Amazing nanostructures. Moscow: Binom Publ., 2011, 209 p.

2. Xiaobin Lu. Simulation of infrared spectra of trace impurities in silicon wafers based on the multiple transmission-reflection infrared method. Scientific Reports, 2021, vol.11, Article number: 1254, 10 p.

3. Tamaz A. Marsagishvili, Jimsher N. Aneli. On Processes of Charge Transfer in Electric Conducting Polymeric Composites. In book: Nanoscience and nanoengineering. Novel applications. Ed. V. B. Dement'ev, A. K. Haghi, V. I. Kodolov, Apple Academic Press, N.Y., 2018. 14p. doi.org/10.1201/9781351138789,

4. McNamara K.M., Scruggs B.E., Gleason K.K. The effect of impurities on the IR absorption of chemically vapor deposited diamond. Thin Solid Films, Elsevier, 1994, vol. 253, issues 1-2, pp. 157-161.

5. Matthew D. McCluskey1. Infrared spectroscopy of impurities in silicon. In: The 6th International Symposium on Advanced Science and Technology of Silicon Materials (JSPS Si Symposium), Nov. 19-23, 2012, Kona, Hawaii, USA, p.51-54.

6. Kamariotis, A.; Boyarkin, O. V.; Mercier, S. R.; Beck, R. D.; Bush, M. F.; Williams, E. R.; Rizzo, T. R. J. Am. Chem. Soc, 2006, vol.128, no. 39, pp. 1259012591.

7. Brown D., Floyd A., Sainsbury M. Spectroscopy of organic substances. Moscow: Mir Publ., 1992, 305 p.

8. Marsagishvili T.A., Machavariani M.N. The role of the effects of frequency and spatial dispersion of the electrolyte in the formation of the IR spectra of adsorbed particles. In book. "Materials of the XIII Frumkin Readings - Double layer and electrochemical kinetics". Tbilisi: "Metsniereba" Publ., 1989, p. 103.

9. Marsagisgvili T., Machavariani M., and Kirillov S. Some aspects of physical and chemical adsorption on surface of amorphous solid. In: Combined and Hybrid Adsorbents. Fundamentals and Applications. Ed. Jose Miguel Loureiro and Mykola T. Kartel. Springer. NATO Security though Science Series. 2006, pp. 349 -356.

10. Stuart, B. Infrared spectroscopy fundamentals and applications; J. Wiley: Chichester, Eng.; Hoboken, N.J., 2004, 224 p.

11. Marsagishvili T., Machavariani M. Theoretical aspects of processes of metal particles adsorption inside channels and on

surface of carbon sorbents. Chemical Problems. 2022. no. 3 (20), pp. 213-222.

12. Prech E., Bulmann F., Affolter K. Determination of the structure of organic compounds. Moscow: Mir Publ., 2006, 439 p.

13. Pentin Yu.A., Vilkov L.V. Physical methods of research in chemistry. Moscow: Mir Publ., 2003, 683 p.

14. Trevor D. Rapson, Helen Dacres. Analytical techniques for measuring nitrous oxide. TrAC Trends in Analytical Chemistry, 2014, vol. 54, pp. 65-74.

15. Kai Töpfer, Debasish Koner, Shyamsunder Erramilli, Lawrence D. Ziegler, Markus Meuwly. Molecular-level understanding of the rovibration spectra of N2O in gaseous, supercritical, and liquid SF6 and Xe. J. Chem. Phys. 2023, vol. 158, pp. 144302-1 -144302-14. doi: 10.1063/5. 0143395.

16. Bolshakov G.F., Glebovskaya E.A., Kaplan Z.G. Infrared spectra and X-ray roentgenograms patterns of heteroorganic compounds. Leningrad: Chemistry Pupl., 1967, 168 p.

ТЕОРЕТИЧЕСКИЕ АСПЕКТЫ КОЛЕБАТЕЛЬНОЙ СПЕКТРОСКОПИИ КОНДЕНСИРОВАННЫХ СИСТЕМ С ПРИМЕСНЫМИ ЧАСТИЦАМИ

Т. Марсагишвили, М. Мачавариани

Тбилисский государственный университет им. И.Иванишвили, Институт неорганической химии и электрохимии им. Р. Агладзе Ул. Миндели, 11, 0186, Тбилиси, Грузия e-mail: tamaz.marsagishvili@gmail com

Аннотация: В работе рассмотрены некоторые проблемы колебательной спектроскопии частиц в конденсированных системах. Один из аспектов теоретических исследований представляет собой изучение колебательных свойств отдельных частиц с учетом нано-размерности окружающих частицу молекул конденсировааной системы. С использованием аппарата температурных функций Грина операторов поляризации конденсированных систем выделены два основных механизма влияния на примесные частицы со стороны среды -сольватационный и флуктуационный. Получены теоретические результаты в рамках этих двух механизмов для расчета изменения колебательного спектра отдельных частиц. Теоретические результаты использованы для анализа экспериментальных данных по колебательным спектрам молекулы N2O в полярных растворителях: метиловом спирте, этиловом спирте, ацетоне, 1.2-дихлорэтане.

Ключевые слова: колебательная спектроскопия, конденсированные системы, сольватация, примесные частицы, пространственная дисперсия, функции Грина.

QARI§IQLARIN HiSSaCiKLaRi iLÖ KONDENSASiYA OLUNMU§ SiSTEMLÖRiN ViBRASiON SPEKTROSKOPiYASININ NÖZaRi ASPEKTLÖRi

T. Marsaqi^vili, M. Ma?avariani

i. ivani§vili adina Tbilisi Dövldt Universiteti R. Aqladze adina Qeyri-üzvi га Elektrokimya institutu Mindeli kügdsi 11, 0186, Tbilisi, Gürcüstan e-mail: tamaz. marsagishvili@gmail. com

Xülasa: Bu i§da kondensasiya olunmu§ sistemlarda hissaciklarin vibrasiya spektroskopiyasinin bazi problemlari nazardan ke?irilir. Nazari tadqiqatin aspektlarindan biri da hissaciyi ahata edan kondensasiya olunmu§ sistemin molekullarinin nanoöl9üsü nazara alinmaqla ayri-ayri hissaciklarin vibrasiya xassalarinin öyranilmasidir. Mühitin hissaciklara tasirinin iki asas mexanizmi farqlandirilib - solvatasiya va fluktuasiya. Ayri-ayri hissaciklarin vibrasiya spektrinin dayi§masini hesablamaq ü9ün bu iki mexanizm 9ar9ivasinda nazari naticalar alda edilib. Nazari naticalar polyar halledicilarda: metanol, etil spirti, aseton va 1,2-dixloretanda N2O molekulunun vibrasiya spektrlari üzra eksperimental malumatlarin tahlili ü9ün istifada olunur.

A?ar sözlar: vibrasiya spektroskopiyasi, kondensasiya olunmu§ sistemlar, halletma, qari§iqli hissaciklar, faza dispersiyasi, Qrin funksiyalari.